Subgap States in Majorana Wires Piet Brouwer Dahlem Center for Complex Quantum Systems Physics...

Subgap States in Majorana Wires

Piet Brouwer

Dahlem Center for Complex Quantum SystemsPhysics DepartmentFreie Universität Berlin

Inanc AdagideliMathias DuckheimDganit MeidanGraham KellsFelix von OppenMaria-Theresa RiederAlessandro Romito

Aachen, 2013

excitations in superconductors

one fermionic excitation → two solutions of BdG equation

Eigenstate of HBdG at e = 0: Majorana fermion,

ue, ve: solution of Bogoliubov-de Gennes equation:

Eigenvalues of HBdG come in pairs ±e, with(particle-hole symmetry)

D: antisymmetric operator

-e

e

Overview• Spinless superconductors as a habitat for Majorana fermions

• Disordered spinless supeconducting wires

• Multichannel spinless superconducting wires

-e

e

• Semiconductor nanowires as a spinless superconductor

• Disordered multichannel superconducting wires

superconductor proximity effect

S N

eh

ideal interface:

Deifrhe(e) = a e-if

reh(e) = a eif

a = e-i arccos(e/D)

n/n

N

e (mV)

x1

x3

x2

Guéron et al. (1996)

x1 x2 x3

S NS

Mur et al. (1996)

I(nA

)S N S

spinless superconductors are topological

scattering matrix for Andreev reflection:

S is unitary 2x2 matrix

S

h

particle-hole symmetry:

combine with unitarity:

Andreev reflection is either perfect or absent

if e = 0

Béri, Kupferschmidt, Beenakker, Brouwer (2009)

e

e

scattering matrix for point contact to S

spinless p-wave superconductors

one-dimensional spinless p-wave superconductor

Majorana fermion end statesbulk excitation gap

Kitaev (2001)

spinless p-wave superconductor

superconducting order parameter has the form

SN D(p)eif(p)rhe

p

Andreev reflection at NS interface

Andreev (1964)

reh-p

*p-wave:

spinless p-wave superconductors

one-dimensional spinless p-wave superconductor

Majorana fermion end statesbulk excitation gap

Kitaev (2001)


superconducting order parameter has the form

SN D(p)eif(p)rhe

reh

p

-p

eih

e-ih

Bohr-Sommerfeld: Majorana bound state if

*

Always satisfied if |rhe|=1.

Proposed physical realizations• fractional quantum Hall effect at ν=5/2

• unconventional superconductor Sr2RuO4 • Fermionic atoms near Feshbach resonance

• Proximity structures with s-wave superconductors, and topological insulators semiconductor quantum well

ferromagnet

metal surface states

Moore, Read (1991)

Das Sarma, Nayak, Tewari (2006)

Gurarie, Radzihovsky, Andreev (2005)Cheng and Yip (2005)

Fu and Kane (2008)

Sau, Lutchyn, Tewari, Das Sarma (2009)Alicea (2010)

Lutchyn, Sau, Das Sarma (2010)Oreg, von Oppen, Refael (2010)

Duckheim, Brouwer (2011)Chung, Zhang, Qi, Zhang (2011)

Choy, Edge, Akhmerov, Beenakker (2011)Martin, Morpurgo (2011)

Kjaergaard, Woelms, Flensberg (2011)

Weng, Xu, Zhang, Zhang, Dai, Fang (2011)Potter, Lee (2010)

(and more)

Semiconductor proposal

Semiconducting wire with spin-orbit coupling, magnetic field

S

N B and SOI

Sau, Lutchyn, Tewari, Das Sarma (2009)Alicea (2010)

Lutchyn, Sau, Das Sarma (2010)Oreg, von Oppen, Refael (2010)

spin-orbit coupling

Zeeman fieldproximity coupling to superconductor


S

N B and SOIspin-orbit coupling

Zeeman fieldproximity coupling to superconductor

p

e


S

N B and SOI

p

e

e

p-pF

BpF

e


S

N B and SOI

p

e

p-pF

BpF

p-pF

BpF

ee


S

N B and SOI

p

e

p-pF

BpF

p-pF

BpF

ee e

p


S

N B and SOI

p

e

p-pF

BpF

p-pF

BpF

ee e

pD



S

N B and SOI

p

e

p-pF

BpF

p-pF

BpF

ee e

pD


B

p

-pF

Mourik et al. (2012)

Spinless superconductor

S

N B and SOI

p

e

p-pF

BpF

p-pF

BpF

ee e

pD


Effective description as a spinless superconductor

Spinless superconductor


e

0

Disorder-induced subgap states


topological phase persists for

Motrunich, Damle, Huse (2001)

e

with disorder:

at critical disorder strength:

density of subgap states:


Disorder-induced subgap states are localized in the bulk of the wire.

Localization length in topological regime: -1

e

n

2lel » xweak disorder

n

e2lel > x~

e

n

2lel < x~almost critical beyond critical

Power-law tail for density of states: if 2lel > x~


e0,max: log-normal distribution e1,min: distribution has algebraic tail near zero energy

Majorana state

fermionic subgap states


Finite L: discrete energy eigenvalues

L

algebraically small energy for large L

exponentially small energy for large L

beyond one dimension

If B D, ap: semiconductor model can be mapped to p+ip model

projection onto “spinless” transverse channels

12321 3

Tewari, Stanescu, Sau, Das Sarma (2012)Rieder, Kells, Duckheim, Meidan, Brouwer (2012)

semiconductor model:

Multichannel spinless p-wave superconducting wire

? ?

L

W

bulk gap:

coherence length

induced superconductivity is weak: and

D

p+ip

Multichannel spinless p-wave superconducting wire

? ?

L

W

bulk gap:

coherence length

induced superconductivity is weak:

Majorana end-states→

and

…Without D’py : chiral symmetry

p+ip

inclusion of py: effective Hamiltonian Hmn for end-states

Hmn is antisymmetric: One zero eigenvalue if N is odd,no zero eigenvalue if N is even.

D

0

Multichannel wire with disorder

? ?

L

W

bulk gap:

coherence length

p+ip


? ?

L

Wp+ip

disorder strength0

Series of N topological phase transitions at

n=1,2,…,N


? ?

L

Wp+ip

Without Dy’: chiral symmetry (H anticommutes with ty)

Topological number Qchiral .

Qchiral is number of Majorana states at each end of the wire.

Without disorder Qchiral = N.

With Dy’:

Topological number Q = ±1

Scattering theory

? p+ip

Without Dy’: chiral symmetry (H anticommutes with ty)

Topological number Qchiral .

Qchiral is number of Majorana states at each end of the wire.

Without disorder Qchiral = N.

With Dy’:

Topological number Q = ±1

N S

L

Fulga, Hassler, Akhmerov, Beenakker (2011)

Chiral limit

? p+ipN S

L

Basis transformation:

Chiral limit

? p+ipN S

L

if and only if

Basis transformation:imaginary gauge field

Chiral limit

? p+ipN S

L


if and only if

imaginary gauge field

Chiral limit

? p+ipN S

L

if and only if

“gauge transformation”

Basis transformation:imaginary gauge field

Chiral limit

? p+ipN S

L



N, with disorder

L

Chiral limit

? p+ipN S

L

N, with disorder

L

: eigenvalues of

Chiral limit

? p+ipN S

L

N, with disorder

L

: eigenvalues of

Distribution of transmission eigenvalues is known:

with , self-averaging in limit L →∞

Series of topological phase transitions

Without Dy’: chiral limit

? ?

L

Wp+ip

topological phase transitions at

n=1,2,…,N

Qchiral

x/(N+1)l

disorder strength


Without Dy’: chiral limit

? ?

L

Wp+ip

With Dy’:


n=1,2,…,N


? ?

L

Wp+ip

With Dy’:


n=1,2,…,N

Dy’

/Dx’

(N+1)l /xdisorder strength

Conclusions• Majorana fermions may persist in the presence of disorder

and with multiple channels

• Disorder leads to fermionic subgap states in the bulk; Density of states has power-law singularity near zero energy.

• Multiple channels may lead to fermionic subgap states at the wire ends.• For multichannel p-wave superconductors there is a

sequence of disorder-induced topological phase transitions. The last phase transition takes place at l=x/(N+1).

disorder strength0

Subgap States in Majorana Wires Piet Brouwer Dahlem Center for Complex Quantum Systems Physics...

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Transcript of Subgap States in Majorana Wires Piet Brouwer Dahlem Center for Complex Quantum Systems Physics...